The contributions to the tail term require retarded boundary conditions, as known from the Feynman–Wheeler
formalism [1]}, [2]} in Quantum Electrodynamics. The Schwinger formalism using in–in states
provides this description.
As well–known, both the usual path–integral formalism [3]}, [4]}, leading to causal Green's
functions with
\(T^*\) –ordering, and, analogously, the so-called in–in formalism, cf. [5]}, [6]},
are exactly defined, also concerning the type of the contributing propagators. The in–in formalism has
also applications in statistical physics, cf. [7]}, [8]}. A critical question concerns
the unitarity of the respective formalism. As has been shown in [9]} this is obeyed for the
in–in formalism at least at two–loop order, the level necessary at 5PN. In the EFT approach the tail diagrams
up 5PN read structurally as shown in Figure REF  using the in–in formalism: these are diagrams
containing three multipole moments with single or double graviton interaction between two in–states.
Also here the gravitons have at most self–couplings
and all end at the worldline being connected to one of the multipole moment insertions. The only contributing diagrams at
5PN are two–loop diagrams. In the action, the 2–loop graviton exchange is integrated out. The necessary Feynman
rules are listed in Appendix . One then may read off the contributions to the conservative tail Lagrangian
(Hamiltonian) from the action directly.
The in–in formalism is described in detail in Appendix .
